Problem: The Chewy Candy Company is test marketing a new candy bar in Seoul and Manila. In Seoul, the company sold $40$ candy bars by the end of the first month, and each month the cumulative total number of candy bars sold increases by $60$. In Manila, the company sold $20$ candy bars by the end of the first month, and the cumulative total number of candy bars sold increases by approximately $80\%$ each month. The company's board assembles at the end of each month to review the cumulative total number of candies sold in each city. What is the first month in which the board will see the cumulative total number of candy bars sold in Manila exceeded the cumulative total in Seoul?
Explanation: Notice that Seoul's total grows linearly while Manila's total grows exponentially. This means Manila's total is bound to exceed Seoul's total at some point. Let's start calculating the cumulative total number of candy bars sold in each place to see when that happens. Month Seoul Manila (Add $60$ each month.) (Multiply by $1.8$ each month.) $1$ $40$ $20$ $2$ $100$ $36$ $3$ $160$ $65$ $4$ $220$ $117$ $5$ $280$ $210$ $6$ $340$ $378$ In conclusion, the cumulative total number of candy bars sold in Manila will first exceed the cumulative total in Seoul in month number $6$.